A Treatise on Ship - Building and Navigation

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'/

t-

"-

'"v- •»

r


-fax 20 x

20"-

be

"c-iz.

^s before.

20^X4

2

X 20 X

4''

3 X

4- 4'

4" 3 X

20 X

I//,

The cube

id.

Three times the fquare of the

= 8000 = 4800 = c6o

4'=

=

4''4- 4'

^ube ot 24

appears, that if any root be divided into

firft

20 X

4 4"^

20x4'

4-

compofed of the four following fums, of the

20' 3 X20'' X

two

=:

parts,

64

13S24

the cube

-lv^.

part. firft

part multiplied,

by the fecond

part.

Three times the

3^^,

firft

part,

multiplied by the fquare of the fecond

part. 4/'Zi,

The cube

of (he fecond part. more than two

If the root confifts of

figures,

the fame

method may

be obferved as in compofing the fquare, by taking the two firft figures for one part, and the next figure for the other part, and fo on till all the This is fo plain, that we think it needfignificant figures are brought in. lefs to give any example here ; and as the only ufe we fhall make of it will be" to fliew how to extrad: the cube root, we lliall juft obferve under this head, that for every figure annexed to the firft figure in the root, there will be three in tlie cube, befides the cube of the firft figure ; thus, 2' is 8, 20' is 8000, 200' is 8000000, &c.

SECT. Evolution

of

II.

^mitities^

is the unfolding, or refolving of any number into the which it is compofed, and is called the extradion of the by means of which, we find a number, that root of any given power

Evolution

parts of

;

being multiplied by

power

itfelf as

many

times

lefs

one, as the

index of the

contains units, will produce the given number.

In

Sect.

EvJt/iiou of

II.

QtTANririE?.

7

In order to this, the method already obferved, iind the fleps taken in the involution ot the binomial root mull: be cartfiiliy att-udtd to, in wiiich it will not be difncult to diJccrn lio.veach part ut' ihe rout is ton-

ctrned in the power.

To

ExtraSl

the

Square Root.

As the fqujre was compofed by multiplication and addition, the lOot muft be found by divifion and lubtraftion. When the root of any number is required, tlie firfl: thing to br; done is to prepare it, by points fet over fuch places as the index of tiie power always beginning at unity, and proceeding towards the lett hand, the given number (which we fhali call the relblvend) be integers, and towards the right hand, in decimal parts ; now the index of the iquare being 2, there muft be a point over every fecond figure, as in the foldiredts, if

lowing example, viz. Let the given number be 5929225 Having pointed the given relolvend

as above,

to find

the

firfl

figure

which in this cafe is 2000, the fquare of which is 4000000. Subtradl this from the refolvend 5929225, and there will remain 1929225, which is called the take the greateft root that

firft

is

contained in the

firft

period,

dividend.

Now

this number contains double the produdl of the firft figure mulby the fecond figure, and the fquare of the fecond figure, as is evident from the method we ufed in compofing the fquare. Divide this dividend, therefore, by double the firft figure, and the

tiplied

when it is multiplied by and the produdl added to the fquare of the fecond figure, the fum does not exceed the dividend. In this example 4000 is double the firft figure in the root, and when this is made a divifor to the dividend, the quotient will be 400. In the next place, multiply this fecond figure by double the firft, or which is the fame thing, by the divifor, and add the fquare thereof to the produdl, and their fum will be 1760000 ; fubtradl this from the dividend, the remainder will be 169225, the fecond dividend. And now we have in effedt fubtraded the fquare of the firft two figures ; for in the firft ftep we fubtradled 2000% and in the next 2x2000 I'he fecond dividend will thereX400-I-400', all which make 2400^ fore contain double the produdt of the firft two figures multiplied by the third, and the fquare of the fame third figure. Therefore, To find the third figure, double the firft two figures for a divifor to quotient will be ihc fecond figure, provided that

double the

firft

figure,

this

Of

8

Involution

and

Chap.

I.

multiply this by the divifor, and add the fquare of the quotient to the produd: ; the fum will be When this is fubtradted from the dividend, the remainder 144900. will be 24325, the third dividend ; to which there mufl be a new divifor found by doubling the figures already found in the root, and the quoAnd by obferving the fame method tient will be 5, the fourth figure. as before in the operation, there will be no remainder, fo that 2435 will this dividend,

be

tlie

and the quotient will be 30

true root required.

Refolvend 5929225 2000^ 4000000

=

1/ divifor 2000 x 2

=

4000)1

;

.

Skct.

Qu A N T

Evolutwi 5/

II.

1

r

I

Ks

Refolvcnd 179109850624(423,'-'^? root

=

16 ifi

2

id dWifor 3(f

843)

=

Afth divifor

5_ 84706)

Slh divifor

847128)

4232-5

843 ^

—

=

8

3

8465 X

576006 508236 6777024 6777024

6

1-7

82x2

3009 2529 480S5

3_ 8465)

divifor

fquarc in

gicattlt

194 164 =:

82)

divifor

5

S4706 X 6

=

847128 X

8

having doubled the root in the quotient how oft it may be had in the dividend; fo as when the quotient figure is annexed to the divifor, and that increafed divifor multiplied by the fame quotient figure, the produdl may be the And fo proceed from greateft number that can be had in the dividend

Obferve always, that

for a divifor,

we

after

are to enquire

:

period to period till the whole is finilhed. By purfuing this method in extrading the root of the fquare number 592922?, the reader will obferve that the operation is exaftly the fame as before, omitting the cyphers.

To Extract

We

fhall obferve the

Cube Root.

the

fime method

have done already in the fquare root

cube

in extracting the ;

that

vided into two parts in different operations,

is,

till

root,

by confidering

we have

it

difcovered

as

wc

as diall

the

fignificant figures thereof.

Let the

number, or refolvend,

whofe

cube

root

is

required be

13824.

When it fift

of two

is

properly pointed as above,

it

appears that the root will con-

figures.

figure in the root will be the greateft root that can be had in period 13000, which is 20 ; the cube of which 8000, muft be fubtradled from the refolvend, and the remainder will be 5824, for a dividend. As we have already fubtradted the cube of the firft part 20, this number muft contain three times the fquare of that firft part multiplied by the fecond part, three times the firft part multiplied by

The

•he

firft

firft

the fquare of the fecond, and the cube of the fecond part, added together.

lo

Jti'vtlution

€>f

and

Chap.

I.

dividend be divided by three times the fquare i^oo, the quetient will be 4, the fecend figure re20^ '^ quired : Then 3 times 4j 3 times 20 x 4% and 4' muft be added together, and the fum fubtradied from the dividend, as in the following

Therefore, then of the firft part

If this

=

example, viz. Refolvend 13824(24 root

=

20' Divifor 3 x

20*=

3x20^x4

= = =

20 X

3 X

4"^

\'

8000 1200) 5824

4800 960 64

5S24

By comparing this example with that which we have given before, where the binomial root 20 -|- 4 is cubed, the reader will obferve, that the very fame numbers which compofe the cube 13824, and are there added together, are here regularly fubtradled from it. This method of extrading the cube root, is, therefore, only the reverfe of that by which it was raifed. We fhall give one example more, without annexing cyphers to the divifors

Let

it

and dividends. be required to extradt the cube root of 948 188 16

94818816(456 I

divifor 4^ x 3 3 X

4"'

— 48

)

30818 dividend

X 5 =1 240.. 300.

3x4x5'= 5'=

2 divifo r 45^

x 3

=

3x45^x6= 3

X45

X 6'

=

6^=

125

"^7125 6075 ) 3693816 dividend 36450.. 4860. 216

3693816 It is here to be noted, that the two laft figures in the dividend muft be excluded, before we enquire how often the divifor (which wants two cyphers) is contained in it, beeaufe when we proceed to find the fecond figure in the root, the firft muft be confidered in the place of tens j and

when

Sect.

Evolution

II.

h

^Quantities.

wlien we are to find the third figure, it muft be confidercd in the place of hundreds, this will appear very plain by annexing the cyphers in the operation.

As root,

the divifor found by this rule, multiplied by the new figure in the is not all that is to be fubftraded from the dividend, but alfo three

times the firfl part multiplied by the Iquare of the fecond, and the cube of the fecond, as in the foregoing example ; it will fomctimes happen that the quotient murt: not" be taken for tlie next figure iii the root, thus, if the dividend 30818 be divided by 4800, the quotient will be 6, but 3X4o'x 6 3X40x6'+6'=33336 which exceeds the dividend* therefore it will be necefiiiry fometimes to try how much the number to be fubftra(fted will amount to by the foregoing rule, before we c^n determine upon the new figure in the root.

+

If,

as

it

often happens,

a

number

has not a root that can be ex-

by a rational number, place as many pairs of cyphers in the fquare, and ternaries of cyphers in the cube, on the right hand of the remainder, as you would have decimal places in the root, and work asbefore, diftingui(hing them from the integers by a comma between ; and thus you may approach infinitely near the exadt root. Mathematicians have proceeded further in the involution and evoluprelTed

tion of quantities, viz. to the 4th, 5th, 6tb, 7th, 8th, and 9th powers, biquadrat, furfolid, fquare cubed, fecond furfolid, and biquadrat

called

fquared, but as

we

not have occafion to apply thefe in pradice, mention them. a given root to any power, or to extra ^'^ ^^^° and 7, but then, becaufe it is the fame as between id

ference

c and 2 betwixt 7 and 5, thefe four numbers not the fame with the difference a the former ranks are faid to be are faid to be in a difcontinued, as Hence the following inferences. continued arithmetical proportion. the fum are in arithmetical proportion continued, J If three quantities the mean, as in this 8 10 ©f the extremes- is equal to the double/of equal to double of the fum of the extremes 8 and 12, is

between is

m

12,

where

20,

'l.'^ffour quantities are of the means.

fo,

the

as 3, 5, 7> 9. here, 3 -?.

If never fo

many

fum of

the extremes

is

equal to the

fum

+ 9 = 12 and + 7 = 12. 5

quantities are fo proportional, the

fum of the ex>

if the number always equal to the double of the middle term, two terms equally diftant of the terms be odd, or to the fum of any from the extremes, as in the following fenes.

tremes

is

2, 4, 6, 8, 10, 12, 1416. 8 X 2 2 -t- 14

or

And

this

4

4- 12

= =8

X 2

= = 16,

&c.

muft always hold good, becaufe the

laft

term comprehends

'

Sect. the

PROPORTION.

II.

together with the

firft,

common

j^

difference fuperaddcd,

as often

as

the number of its place is diftant from the firft term But the firft term has no addition of the difference at all ; and as the fecond term has one difference or ratio more than the firft ; the third one more than the fecond, 6cc. fo the laft but one has one lefs than the laft of all ; the laff but two one lefs than the laft but one, &c. whence the fum of any two of thefe equally diftant from the extremes muft be equal to the fum of the extremes, becaufe one increafes as much as the other decreafes. Therefore, the fum of any number of terms in fuch a progreffion may be had, by multiplying the fum of the extremes by half the number :

of the terms.

To find the fum of never fo many quantities in this progrefilon, it is only neceffary that the extremes and the number of terms be given fo that if by having the firft term and the common excefs you would find the laft, it might be done with great difpatch, by multiplying the number of terms, leffened by unity, into the common excefs, and then adding the firft term to the product. :

Thus,

term of a progreffion of 73 places were required, and were 4, and the firft term 3 ; you need only multiply 72 by 4, and to the produdl 288 add 3, and you have 291 for tlie laft term in the progrefilon. So that if the progreffion begins with a cypher, which is the rnoft natural and fimple of all, then the fum of all the terms will be equal to the fum of the extremes multiplied by half the number of the terms. the

if

the

common

Thus

laft

difference

fuppofe Qj 3> 6' 9> 12, 15, 18, 21, 24, 27, 30, 33, 36, 29.

The

term 39, multiplied by 14, the whole number of terms gives 546; the half of which 273, is the fum of all the terms. From whence it will follow, that the fum of all the terms in any fuch progreffion beginning from o, is half the fum of fo many terms, all e_

laft

qual to the greateft.

SECT.

II.

0/ Geometrical Proportion. GEometrical

proportion is when in comparing two or more numbers, divided by the other, the quotient is called the ratio ; and feveral ratio's are equal, the numbers are faid to be in a geometrical

one

when

is

C2

pro-

PROPORTION.

r4

Chap.

II,

proportion ; thu?, 2 : 6 : 5: 15 are proportionals ; for 6 divided by 2 is 3, and I 5 divided by 5 is 3, and fo tiie ratio's are equal. When two numbers are compared, the former is called the antecedent, and the hitter the confequent-, Inst v/hen more than two are compared, they are called terras, of which the firft and lafl: are called extremes, and all the intermediate ones, nicai>s. In order to know whether numbers be proportionals, it is only finding the ratio of each pair, and here it will be indifferent whether the ante-, cedents or confequents of every pair be made divifors, as in the preceding numbers 2 ; 6 : : 5 : 15., if 2 the antecedent be divided by 6, the conle:

5 be divided by 15, the quotient is 4. fo the antecedents were made divifors. When in a rank of numbers they increafe in a geometrical proportion, the ratio will be a common multiplier, and is found by dividinganyoneof the confequeiit terms by its antecedent, for fo the quotient will be the ratio. quent, the quotient tlie ratio's

is

\,

and

if

are equal, as before

when

H^rc the common multiplier is A ^ J 3' 9> ^7» ^^* ^43» ^^"[2,4, 8, i6y 32, &c. Here the common multiplier is It

ratio,

is

plain that if either

produd

the

When

will be

3. 2.

of thefe antecedent terms be muhiplied by the its

confequent.

numbers theydecreafe in a geometrical proportion, the ratio will be a common divifor, and is found by dividing any one of the antecedent terms by its confequent. a in rank of

y 243, 81, 27, 9, I 32, 16, 8, 4,

3, 2,

&c. Here the common &c. Here the common

divifor

is

3.

divifor

is

2.

In like manner, if either of thefe terms is divided by the ratio, quotient will be the next term in the progreffion. If in comparing feveral numbers together, we find the ratio of the

tlie

firfl

and fccond, to be the fame with the ratio of the fecond and third, third and fourth, fourth and fifth, and fo on ; thofe numbers are in a geometrical

proportion continued,

.^

numbers together, we find the ratio of the firft and fccond to be the fame as the ratio of the third and fourth, but that thofe numbers are the ratio of the fecond and third is not the fame called difcontinued proportionals, as 2: 4:: 6: 12, for the progrelTion If in comparing four

;

llops here at 4.

The manner of exprefling continued proportionals is by feparating the terms by two points, as 2:4:8: 16 32, &c. but in difcontinued pro**, portionals the terms where the progreiTion Hops are feparated by four :

points, as

5

:

10

:

:

6

:

12

:

:

7

:

14,

ori4

:

7

:

:

12

:

6

:

:

10

:

5.

PROPORTION,

Sect. H,

PROPOSITION

1.

If three numbers are in a geometrical proportion,

the product of the extremes will be equal to the piodu£t of the middle term multiplied into itfelf, or which is the fame thing, to the fquare of the middle term.

As

in thefe

numbers

2x18

6,

18,

= 6x6 =

2^'

2,

PROPOSITION

II.

If four numbers are in geometrical proportion, (whether continued or difcontinued) the produdt of the extremes will be equal to the produifl

of the z means. Let the numbers be 5

ro

:

:

:

6

:

r

2

5XI2=:l0x6=;6o 10

or 5 5 X 40 :

:

20

:

40

= 10 K 20 = 209

tl:iefe tv/o propofitions the following inferences may be drawn, viz, If the produdl of any two numbers is equal to the fquare of a third, thofe three numbers are in geometrical proportion continued. 2d, If the produdt of any two numbers is equal to the product of any other two, thofe four numbers are proportionals, and the numbers multiplied into each other will be either 2 means, or 2 extremes, as in the Allowing examples, viz.

From ijfy

=

2 X i6r=4x 8 32 16 4 :: 8:2 4 16 :: 2:8, &:c. :

:

or

8 X i2

= 6x 16 = 96

:6:: 16 6: 12 :: 8 8

:

12

:

16,

&c

PROBLEM To

I.

mean proportional between any two given numbers, Note^ By a mean proportional we are to underftand fuch a number as if multiphed by itfelf, the produd will be equal to the produd: of the two given numbers. Rule, Multiply the given numbers by one another, and extrad the fquare root of the produd, that root will be the mean required. Exfind a

PROPORTION.-

1^

Example, Let the given numbers be 3 X

27

= 81

the fquare root of

3

which

PROBLEM To ratio

Chap.

II;

and 27. is

9 the

mean

required.

n.

numbers, fo that the be equal to the ratio of the firft and

find a fourth proportional to the three given

of the third and fourth

may

fecond. Rule,

Multiply the fecond number by the third, and divide the produci by the firlt, tlae quotient will be the fourth number required.. Example', Let the given numbers be 2 6 7. :

6

y.

J = 42,

and 42 -r

2

:

:

= 21, the fourth number

required.

is evident from the two foregoing prawhere there are three numbers given to find a fourth, though, the fourth is not known, we know that the produdl of it when multiplied by the firfl will be equal to the produdt of the fecond and third numbers, (per prop. 11.) therefore, if that produd: is divided by the firfl This is the term, the quotient will be the fourth number required.

The

reafon of both thefe rules

pofitions, for

foundation of the rule of three,

which we fuppofe

the reader acquainted

with already. In finding a fourth proportional where three numbers are given, the firfl give the ratio, and the queflion as to the fourth proportional

two

concerns the third number. Dired: proportion is, when the greater the term is by which the queftion is made, the fourth term will be alfo the greater j and the leffer that term is, the fourth will alio be the lefiTer. Reciprocal^ or inverfe proportion is, when the greater the term is by which the queflion is made, the fourth will be leffer, and the lefler that is, the greater the fourth. Continual proportion, thus exprefTed, 44-, is, when all the terms between the firfl and the lafl are both antecedents and confequents in the

term

fame proportion. Example, 8, 12,

iS, 27,

are -H-;

for 8

:

12

:

:

12

:

18

:

:

18

:

27.

"Wherefore in fueh feries, the lafl term fubtradted from the fum of all the terms will give the fum of all the antecedents, and the firfl term fubtrafted from the laid fum will give the fum of all the confequents. If four quantities be proportional, they will alfo be fo alternately, inverfely, in compofition, in divifion,. converfely,

and mixtly.

Ex-

Sx:c cT. II.

p R

r

R r

I

N.

PROPORTION.

rS

As

m

I

is

to the multiplier, fo

divifion, as the divilbr

is

Chap.

IF.

the multiplicand to the produdl ; and fo is the dividend to the quotient..

is

to i,

P R O P O

.9

I

T

I

O

N.

V.

two numbers be multiplied, or divided by any fame third number, the produfts or quotients will be proportional to the numbers fo mulLet the numbers be 2 and 4, each to be multiplied tiplied or divided. 6 1 2 or if 1 8 and 1 2, and 2 : 4 then 3x2=6, and 3 x 4 by 3 6 5^. 15 be divided each by 3, the quotients will be 6 and 5, and 18: 1 5 If any

=

J

:

:

:

;

: :

The powers

:

or roots of proportionals will likewife be propoitionals,

6 root 2 4 3 4: 16 9 36 fquare 8:64:: 27:216 cube ;

:

:

:

:

:

:

we have confidered the dodrine of proportion, only with renumbers ; but as any quantity may be reprefented by numbers,

Hitherto

iped

to

has been faid with regard to them may hkewife be applied to anyline of 2 feet long has thing that can be augmented or diminiflied. the fame proportion to a line of 6 feet long that the number 2 has to 6, but the method of finding the proportions of lines, &c. to one another requires the knowledge of the principles of geometry,. wJiich fball all that

A

be the fubjed of the next chapter.

CHAR

HA

C

Of

As

SECT.

IIL

P.

I^

GEOMETRY, we

treat of numbers by comparing tlicm one another, without conlidering their relation to any geometry we Ihcw how to compare quantities to one another, and find their proportions without a-

in arithmetick

to

particular quantity, lb in

rithmetical calculation.

Geometry may therefore be fitly called, The fcience of e:^-. tenfion abftra^edly confidered, without any regard to matter.

Geometrical

Definitions,

Quantity is any thing that can be augmented or dimil^ef. nifhed, and may comprehend extenfion, weight, motion, ficc. for one may be taken as greater or lefler, heavier, or lighter, I.

or flower, in relation to another, of things of the fame kind ; but there can be no comparifon between quantities of different kinds ; as hours and miles ; for an hour is neither Iwifter,

greater nor

lefs, heavier nor lighter, fee. than a mile. All things that are capable of extenfion are to be confulered either as lines, furfaces, or folids.

Def\

Dcf,

2.

A

3.

length only

is

line

is

a quantity of one dimenfion,

where the

confidered.

A

furface is a quantity confidered under two dimen* Def. 4, viz length and breadth. DeJ. 5. A folid is that which has three dimenfions, 'ciz. lengtJi, breacTth, and thicknefs, thefe two lall arc fometimes called height and depth. DcJ\ 6. A point, in the mathematical fenfe, and in refpc6l of continual quantity, is that wherein neither of the foregoing diHcienfions are confidered. It therefore confifts of no parts ; for then it would be a folid, furface, or line. It is analogous to an inftant in time, which partakes neither of the part or "the future. Tiie centers of circles, 6cc. in diagrams are not mathematical points, but fenfible objeds whereby the underftanding, confiderfions,

ing them abftraftedly,

is

affifled in

13

mathematical fpccuhtions. Def.

20 Plate 1.

F'g-

I-

G E

M

E rR

Chap.

r.

III.

D another,as. figure, whofe fides and angles are all equal, is called a geome-

Fig.

cj.

A

trical fquare.

Dcf. 21.

A

r»o right angle,

and

rhombus

is

a figure that hath four equal fides, but Tig 10.

the oppofite angles being equal,

G E F= G

H F,

but

all

oblique.

i^iz.

EC H=L'F H,

;

GEOMETRK

24 PlateI-

Fig-

I ^-

Fit?. 17. *

Fig.

I 7.

Chap.

III.

A

right

redtangle, or aright angled parallelogram, hath four angles, and its oppofite fides equal and parallel, viz, L ; this figure is often called an obL, and I

long,

or long fquare.

Def. 22.

IM = k

K=M

A'/' 23. A rhomboides is an oblique-angled parallelogram, and the fides that form the angles are unequal. J^^f' 24- Every quadrilateral figure that has neither oppofite fides,

A

nor oppofite angles equal, right line

gure to

its

is

called a trapezium.

drawn from any angle

oppofite angle

(B),

is

(as

D)

called a

in a

four fided

diagonal,

fi-

and di-

two triangles (A B D and BCD). Def. 25. A polygon is a figure that hath more than four fides, and may be either regular or irregular. Def. 26. A regular polygon has all its fides and angles equal, and may be infcribed in a circle, and all the angular points (a^ ^j C-, d, e,f) will touch the circumference. Regular polygons derive their names firom the number of their fides or angles at the center of the circle they are infcribed in thus a polygon of 5 fides is called a pentagon, of 6 fides a hexagon, of 7 fides a heptagon, of 8 fides an odlagon, &c. I^ef- 27. An irregular polygon has many unequal fides, ftandE F G. ing at unequal angles, as A B C All irregular polygons may be reduced to regular figures, by drawing diagonal lines in them ; thus the polygon AB C D E FG, B, B F, F C, and C E being drawn, will be the diagonals reduced to five triangles, 'uiz. A B G, G B F, B F C, F C E, and vides the figure into

Fig. 14.

Fig. ir. ^

D

G

CDE. S

E

C

T.

II.

GEOMETRICAL PROPOSITIONS; PROPOSITION Fig. 16.

\'^^ P^^^ P°'"' f\^

^" ^

equal to a givea one,

Ooen your

'^S^^^ ^"^^*

I.

Problem.

(^^ ")*

*° "^^^* an angle

(A B C.)

compaflfes to any convenient diftance,

and from

B as

M

G E E R T. 25 anarch, asj/j with the fame extent de- Plate I. fcribe the arch g, from the center ; then take the chord s t your compalTes, and fet it off fromy" to k in the arch/^, and k, and fo the angles ABC, and k oJ\ will be draw the line Sect.

IT.

T

a center, defcrlbe

as

f

m

equal.

Hence an

angle of any number of degrees may be made, for with the radius of any divided circle we defcribe an arch (as / m) from the point C, as center, in a right line, as C B, and then take the quantity of the given angle (fuppofe 20 degrees) from the divided circumference of that circle by whofe radius the arch was defcribed, and fet that off upon the arch from / to w, I will be the angle required; this and draw the line Cm, in effedt is making one angle equal to another given one, for any angle may be found in the divided circle by drawing two femidiameters to two points of divifion, including the meafure if

3"*

mC

thereof in the circumference. If the angle is already made,- defcribe an arch as before to meet both fides of the angle, the extent of this arch meafured upon the divided circumference will give the quantity of the angle.

PROPOSITION

II.

Theorem.

two

right lines interfeft or cut one another, the oppofitepjcr. ly, angles will be equal ; the contiguous angles, as ^ and c, taken toge-

If

make 180 degrees (by Inf. 3,!)^/. 16), but if inflead of fum of the angles a and d will likewife be 1 80 degrees, therefore c and d muft be equal ; for the fame reafon, as the angles c or ^, added to the angles a or I', will make 180 dether will

c

we

take d, the

grees,

To

it

will appear that thofe oppolite angles are likewife equal.

by numbers, let the angle